Quantum stochastic Feynman rules and extended Wigner statistics

We introduce a Feynman diagram calculus for quantum stochastic fields and show that from the weak coupling limit of a Bose quantum field reservoir in a Gaussian state one obtains a quantum stochastic field process which has extended Wigner statistics. That is, we calculate explicitly the moments using the diagram rules and show that only moments corresponding to non-crossing diagrams are non-trivial. We further give the physical mechanism underlying this fact; namely the combination of momentum conservation (arising from the correct treatment of the full responsive interaction) and the mass-shell energy conservation for the virtual (stochastic) reservoir quanta. In our analysis we introduce a volume cut-off initially for the system particle to discretize its energy spectrum and then take the weak coupling limit. The individual noise fields associated with energy state transitions were Gaussian however, when the volume cut-off is then removed and we recover the noise correlations. Thus for this problem the removal of the cut-off and the weak coupling limit are in fact interchangeable.